C₅×#12684 #11935404

order	1280
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#12684

Subgroups

	subgroup	G/H
center	C₅×C₈	#120
commutator	C₂³	C₂²×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1779
prime radical	C₅×#585	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₅×C₈ ◁ C₂²×C₅×C₈ ◁ G

894 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 894 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2,3,4,5)(6,7,10,15,11,16,25,35,12,17,26,36,27,37,48,58)(8,13,20,30,21,31,44,54,19,29,42,52,43,53,63,67)(9,14,23,33,24,34,47,57,22,32,45,55,46,56,64,68)(18,38,40,59,41,60,62,69,39,28,49,50,61,51,65,66);(6,8)(7,13)(9,18)(10,20)(11,21)(12,19)(14,28)(15,30)(16,31)(17,29)(22,39)(23,40)(24,41)(25,44)(26,42)(27,43)(32,38)(33,50)(34,51)(35,54)(36,52)(37,53)(45,49)(46,61)(47,62)(48,63)(55,59)(56,60)(57,66)(58,67)(64,65)(68,69);(6,9)(7,14)(8,18)(10,23)(11,24)(12,22)(13,28)(15,33)(16,34)(17,32)(19,39)(20,40)(21,41)(25,47)(26,45)(27,46)(29,38)(30,50)(31,51)(35,57)(36,55)(37,56)(42,49)(43,61)(44,62)(48,64)(52,59)(53,60)(54,66)(58,68)(63,65)(67,69)

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5672);